Question: Solve for $x$, ignoring any extraneous solutions: $\dfrac{x^2 - 12}{x + 7} = \dfrac{-10x - 33}{x + 7}$
Answer: Multiply both sides by $x + 7$ $ \dfrac{x^2 - 12}{x + 7} (x + 7) = \dfrac{-10x - 33}{x + 7} (x + 7)$ $ x^2 - 12 = -10x - 33$ Subtract $-10x - 33$ from both sides: $ x^2 - 12 - (-10x - 33) = -10x - 33 - (-10x - 33)$ $ x^2 - 12 + 10x + 33 = 0$ $ x^2 + 21 + 10x = 0$ Factor the expression: $ (x + 3)(x + 7) = 0$ Therefore $x = -3$ or $x = -7$ However, the original expression is undefined when $x = -7$. Therefore, the only solution is $x = -3$.